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Quantum conformal superspace

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Abstract

For a compact connected orientablen-manifoldM, n ≥ 3, we study the structure ofclassical superspace \(\mathcal{S} \equiv {\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} \mathcal{D}}} \right. \kern-\nulldelimiterspace} \mathcal{D}}\),quantum superspace \(\mathcal{S}_0 \equiv {\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} {\mathcal{D}_0 }}} \right. \kern-\nulldelimiterspace} {\mathcal{D}_0 }}\),classical conformal superspace \(\mathcal{C} \equiv {{\left( {{\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} \mathcal{P}}} \right. \kern-\nulldelimiterspace} \mathcal{P}}} \right)} \mathord{\left/ {\vphantom {{\left( {{\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} \mathcal{P}}} \right. \kern-\nulldelimiterspace} \mathcal{P}}} \right)} \mathcal{D}}} \right. \kern-\nulldelimiterspace} \mathcal{D}}\), andquantum conformal superspace \(\mathcal{C}_0 \equiv {{\left( {{\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} \mathcal{P}}} \right. \kern-\nulldelimiterspace} \mathcal{P}}} \right)} \mathord{\left/ {\vphantom {{\left( {{\mathcal{M} \mathord{\left/ {\vphantom {\mathcal{M} \mathcal{P}}} \right. \kern-\nulldelimiterspace} \mathcal{P}}} \right)} {\mathcal{D}_0 }}} \right. \kern-\nulldelimiterspace} {\mathcal{D}_0 }}\). The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S 0,C, andC 0 areilh orbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS 0 and quantum conformal superspaceC 0 are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS 0 andC 0. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC 0 will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.

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References

  1. Arnowitt, R., Deser, S., and Misner, C. (1962). InGravitation: An Introduction to Current Research, L. Witten, ed. (John Wiley & Sons, New York).

    Google Scholar 

  2. Balachandran, A. P. (1989). InGeometrical and Algebraic Aspects of Nonlinear Field Theory, S. De Filippo, M. Marinaro, G. Marmo, and G. Vilasi, eds. (Elsevier Science Publishers).

  3. Bröcker, T., and Jänich, K. (1982).Introduction to Differential Topology (Cambridge University Press, Cambridge).

    Google Scholar 

  4. Bruhat, Y. (1962). InGravitation: An Introduction to Current Research, L. Witten, ed. (John Wiley & Sons, New York)

    Google Scholar 

  5. Choquet-Bruhat, Y., and York, J. (1980). InGeneral Relativity and Gravitation, A. Held, ed. (Plenum Press, New York), vol. I.

    Google Scholar 

  6. Davis, M., and Morgan, J. (1934). InThe Smith Conjecture, J. Morgan and H. Bass, eds. (Academic Press, New York).

    Google Scholar 

  7. Ebin, D. (1970).Proc. Symp. Pure Math., Amer. Math. Soc. 15, 11.

    Google Scholar 

  8. Fischer, A. (1970). InRelativity, M. Carmeli, S. Fickler, and L. Witten, eds. (Plenum Press, New York).

    Google Scholar 

  9. Fischer, A., and Marsden, J. (1977).Canadian J. Math. 1, 193.

    Google Scholar 

  10. Fischer, A., and Marsden, J. (1979). InProc. International School of Physics “Enrico Fermi,” LXVII — Isolated Gravitating Systems in General Relativity, J. Ehlers, ed. (North-Holland, Amsterdam/New York).

    Google Scholar 

  11. Fischer, A., and Moncrief, V. (1994) InPhysics on Manifolds. (Proc. International Colloquium in Honour of Yvonne Choquet-Bruhat), M. Flato, R. Kerner, and A. Lichnerowicz, eds. (Kluwer Academic Publishers, Boston).

    Google Scholar 

  12. Fischer, A., and Moncrief, V. (1994). InProc. Cornelius Lanczos International Centenary Conference, J. Brown, M. Chu, D. Ellison, and R. Plemmons, eds. (Society for Industrial and Applied Mathematics, Philadelphia).

    Google Scholar 

  13. Fischer, A., and Moncrief, V. (1996).Gen. Rel. Grav. 28, 209.

    Google Scholar 

  14. Fischer, A., and Moncrief, V. (1995). InGlobal Structure and Evolution in General Relativity, S. Cotsakis and G. W. Gibbons (Springer-Verlag, Heidelberg).

    Google Scholar 

  15. Fischer, A., and Moncrief, V. (1995). “Reduction of the Einstein equations in (3+1)-dimensions to a Hamiltonian system on the cotangent bundle of Teichmüller space”, to appear.

  16. Fischer, A., and Tromba, A. (1984).Mathematische Annalen 267, 311.

    Google Scholar 

  17. Fischer, A., and Tromba, A. (1984).J. für die reine und angewandte Mathematik 352, 151.

    Google Scholar 

  18. Friedman, J. (1990). InConceptual Problems in Quantum Gravity, A. Ashtekar and J. Stachel, eds. (Birkhäuser, Boston).

    Google Scholar 

  19. Friedman, J., and Witt, D. (1986).Topology 25, 35.

    Article  Google Scholar 

  20. Freedman, M., and Yau, S. T. (1983).Topology 22, 179.

    Article  Google Scholar 

  21. Gribov, V. (1976).Nucl. Phys. B 139, 1.

    Google Scholar 

  22. Hatcher, A. (1976).Topology 15, 343.

    Article  Google Scholar 

  23. Hempel, J. (1976).Annals of Mathematics Studies, Number 86 (Princeton University Press, Princeton, N. J.).

    Google Scholar 

  24. Hsiang, W. (1967).Bull. Amer. Math. Soc. 73, 55.

    Google Scholar 

  25. Hsiang, W. (1967).Ann. Math. 85, 351.

    Google Scholar 

  26. Hsiang, W. (1971).Tamkang Journal of Mathematics, Tamkang College of Arts and Sciences, Taipei 73, 1.

  27. Jaco, W. (1980).Lectures on three-manifold topology, (Conference Board of the Mathematical Sciences, 43, A.M.S., Providence, R.I.).

    Google Scholar 

  28. Kneser, H. (1929).Jber. Deutsch. Math.-Verein 38, 248.

    Google Scholar 

  29. Lichnerowicz, A. (1944).J. Math. Pures Appl. 23, 37.

    Google Scholar 

  30. Mess, G. (1995). “Homotopically trivial symmetries of 3-manifolds are almost always toral”, to appear.

  31. Milnor, J. (1961).Amer. J. Math. 1, 7.

    Google Scholar 

  32. Milnor, J. (1956).Ann. Math. 63, 272.

    Google Scholar 

  33. Milnor, J. (1956).Ann. Math. 63, 430.

    Google Scholar 

  34. Mitter, P., and Viallet, C., (1981).Commun. Math. Phys. 79, 455.

    Article  Google Scholar 

  35. Moncrief, V. (1989).J. Math. Phys. 30, 2907.

    Article  Google Scholar 

  36. Moncrief, V. (1990).J. Math. Phys. 31, 2978.

    Google Scholar 

  37. Omori, H. (1970).Proc. Symp. Pure Math., Amer. Math. Soc. 15, 167.

    Google Scholar 

  38. Orlik, P. (1972).Seifert manifolds (Lecture Notes in Mathematics 291, Springer-Verlag, New York).

    Google Scholar 

  39. Orlik, P., and Raymond, F. (1968). InProc. Conference on Transformation Groups (New Orleans, 1967), P. Mostert, ed. (Springer-Verlag, New York).

    Google Scholar 

  40. Raymond, F. (1968).Trans. Amer. Math. Soc. 131, 51.

    Google Scholar 

  41. Satake, I. (1956).Proc. Nat. Acad. Sci. USA 42, 359.

    Google Scholar 

  42. Scott, P. (1983).Bull. London Math. Soc. 15, 401.

    Google Scholar 

  43. Singer, I. (1978).Commun. Math. Phys. 60, 7.

    Article  Google Scholar 

  44. Sorkin, R. (1989). InGeometrical and Algebraic Aspects of Nonlinear Field Theory, S. De Filippo, M. Marinaro, G. Marmo, and G. Vilasi, eds. (Elsevier Science Publishers).

  45. Spanier, E. (1966).Algebraic Topology (McGraw-Hill Book Company, New York).

    Google Scholar 

  46. Thurston, W. (1978).The geometry and topology of 3-manifolds Lecture notes from Princeton University, Princeton, N.J.

  47. Thurston, W. (1982). InLow-dimensional topology, London Mathematical Society Lecture Note Series 48, edited by R Brown and T L Thickstun (Cambridge University Press, Cambridge).

    Google Scholar 

  48. Thurston, W. (1982).Bull. Amer. Math. Soc. 6, 357.

    Google Scholar 

  49. Witt, D. (1986).J. Math. Phys. 27, 573.

    Article  Google Scholar 

  50. Waldhausen, F. (1967).Topology 6, 505.

    Article  Google Scholar 

  51. Waldhausen, F. (1968).Ann. Math. 18, 56.

    Google Scholar 

  52. Wolf, J. A. (1977).Spaces of Constant Curvature (3rd. ed., Publish or Perish Press, Berkeley, California).

    Google Scholar 

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Fischer, A.E., Moncrief, V. Quantum conformal superspace. Gen Relat Gravit 28, 221–237 (1996). https://doi.org/10.1007/BF02105425

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